Let C_1, C_2 be two circles touching each oth | vedclass

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Question

(a)

Given,

$A B$ is diameter of circle $C_1$.

$B A_1=2$

$B A_2=3$

$B A_3=4$

Let radius of circle $C_1=r_1$ and radius of circle $C_2

Vedclass · Accepted Answer

$\frac{\sqrt{30}}{5}, \frac{3 \sqrt{30}}{10}$

Vedclass · Answer

(a)

Given,

$A B$ is diameter of circle $C_1$.

$B A_1=2$

$B A_2=3$

$B A_3=4$

Let radius of circle $C_1=r_1$ and radius of circle $C_2=r_2$

$	herefore B A=2 r_1 	ext { and } A C=2 r_2$

$\Rightarrow B M=\frac{1}{2} B A_1=1$

$\Rightarrow B N=B A_2+\frac{1}{2} A_2 A_3$

$=3+\frac{1}{2}=\frac{7}{2}$

In $	riangle B M P$ and $	riangle B N Q$

$\frac{B M}{B N} =\frac{B P}{B Q}$

$\frac{1}{7 / 2} =\frac{r_1}{2 r_1+r_2}$

$2 r_2 =3 r_1$

Now, $\quad B A_2 	imes B A_3=B A 	imes B C$

$3 	imes 4=2 r_1\left(2 r_1+2 r_2ight)$

$12=4\left(r_1^2+r_1 r_2ight)$

$r_1^2+r_1 r_2=3$

From Eqs.$(i)$ and $(ii)$, we get

$r_1=\sqrt{\frac{6}{5}}=\frac{\sqrt{30}}{5}$ and $r_2=\frac{3 \sqrt{30}}{10}$

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